Non-Noether symmetries in Hamiltonian Dynamical Systems

Symmetries play essential role in dynamical systems, because they usually simplify analysis of evolution equations and often provide quite elegant solution of problems that otherwise would be difficult to handle. In Lagrangian and Hamiltonian dynamical systems special role is played by Noether symmetries — an important class of symmetries that leave action invariant and have some exceptional features. In particular, Noether symmetries deserved special attention due to celebrated Noether's theorem, that established correspondence between symmetries, that leave action functional invariant, and conservation laws of Euler-Lagrange equations. This correspondence can be extended to Hamiltonian systems where it becomes more tight and evident then in Lagrangian case and gives rise to Lie algebra homomorphism between Lie algebra of Noether symmetries and algebra of conservation laws (that form Lie algebra under Poisson bracket).

Role of symmetries that are not of Noether type has been suppressed for quite a long time. However, after some publications of Hojman, Harleston, Lutzky and others (see [16], [36], [39], [40], [49]-[57]) it became clear that non-Noether symmetries also can play important role in Lagrangian and Hamiltonian dynamics. In particular, according to Lutzky [51], in Lagrangian dynamics there is definite correspondence between non-Noether symmetries and conservation laws. Moreover, each generator of non-Noether symmetry may produce whole family of conservation laws (maximal number of conservation laws that can be associated with non-Noether symmetry via Lutzky's theorem is equal to the dimension of configuration space of Lagrangian system). This fact makes non-Noether symmetries especially valuable in infinite dimensional dynamical systems, where potentially one can recover infinite sequence of conservation laws knowing single generator of non-Noether symmetry.

Existence of correspondence between non-Noether symmetries and conserved quantities raised many questions concerning relationship among this type of symmetries and other geometric structures emerging in theory of integrable models. In particular one could notice suspicious similarity between the method of constructing conservation laws from generator of non-Noether symmetry and the way conserved quantities are produced in either Lax theory, bi-Hamiltonian formalism, bicomplex approach or Lenard scheme. It also raised natural question whether set of conservation laws associated with non-Noether symmetry is involutive or not, and since it appeared that in general it may not be involutive, there emerged the need of involutivity criteria, similar to Yang-Baxter equation used in Lax theory or compatibility condition in bi-Hamiltonian formalism and bicomplex approach. It was also unclear how to construct conservation laws in case of infinite dimensional dynamical systems where volume forms used in Lutzky's construction are no longer well defined. Some of these questions were addressed in papers [11]-[14], while in the present review we would like to summarize all these issues and to provide some examples of integrable models that possess non-Noether symmetries.

Review is organized as follows. In first section we briefly recall some aspects of geometric formulation of Hamiltonian dynamics. Further, in second section, correspondence between non-Noether symmetries and integrals of motion in regular Hamiltonian systems is discussed. Lutzky's theorem is reformulated in terms of bivector fields and alternative derivation of conserved quantities suitable for computations in infinite dimensional Hamiltonian dynamical systems is suggested. Non-Noether symmetries of two and three particle Toda chains are used to illustrate general theory. In the subsequent section geometric formulation of Hojman's theorem [36] is revisited and some examples are provided. Section 4 reveals correspondence between non-Noether symmetries and Lax pairs. It is shown that non-Noether symmetry canonically gives rise to a Lax pair of certain type. Lax pair is explicitly constructed in terms of Poisson bivector field and generator of symmetry. Examples of Toda chains are discussed. Next section deals with integrability issues. An analogue of Yang-Baxter equation that, being satisfied by generator of symmetry, ensures involutivity of set of conservation laws produced by this symmetry, is introduced. Relationship between non-Noether symmetries and bi-Hamiltonian systems is considered in section 6. It is proved that under certain conditions, non-Noether symmetry endows phase space of regular Hamiltonian system with bi-Hamiltonian structure. We also discuss conditions under which non-Noether symmetry can be "recovered" from bi-Hamiltonian structure. Theory is illustrated by example of Toda chains. Next section is devoted to bicomplexes and their relationship with non-Noether symmetries. Special kind of deformation of De Rham complex induced by symmetry is constructed in terms of Poisson bivector field and generator of symmetry. Samples of two and three particle Toda chain are discussed. Section 8 deals with Frölicher-Nijenhuis recursion operators. It is shown that under certain condition non-Noether symmetry gives rise to invariant Frölicher-Nijenhuis operator on tangent bundle over phase space. The last section of theoretical part contains some remarks on action of one-parameter group of symmetry on algebra of integrals of motion. Special attention is devoted to involutivity of group orbits.

Subsequent sections of present review provide examples of integrable models that possess interesting non-Noether symmetries. In particular section 10 reveals non-Noether symmetry of $n$-particle Toda chain. Bi-Hamiltonian structure, conservation laws, bicomplex, Lax pair and Frölicher-Nijenhuis recursion operator of Toda hierarchy are constructed using this symmetry. Further we focus on infinite dimensional integrable Hamiltonian systems emerging in mathematical physics. In section 11 case of Korteweg-de Vries equation is discussed. Symmetry of this equation is identified and used in construction of infinite sequence of conservation laws and bi-Hamiltonian structure of KdV hierarchy. Next section is devoted to non-Noether symmetries of integrable systems of nonlinear water wave equations, such as dispersive water wave system, Broer-Kaup system and dispersiveless long wave system. Last section focuses on Benney system and its non-Noether symmetry, that appears to be local, gives rise to infinite sequence of conserved densities of Benney hierarchy and endows it with bi-Hamiltonian structure.

The basic concept in geometric formulation of Hamiltonian dynamics is notion of symplectic manifold. Such a manifold plays the role of the phase space of the dynamical system and therefore many properties of the dynamical system can be quite effectively investigated in the framework of symplectic geometry. Before we consider symmetries of the Hamiltonian dynamical systems, let us briefly recall some basic notions from symplectic geometry.

The symplectic manifold is a pair $(M,\omega )$ where $M$ is smooth even dimensional manifold and $\omega$ is a closed $$d\omega =0$$ and nondegenerate 2-form on $M$. Being nondegenerate means that contraction of arbitrary non-zero vector field with $\omega$ does not vanish $$i_X\omega =0\iff X=0$$ (here $i_X$ denotes contraction of the vector field $X$ with differential form). Otherwise one can say that $\omega$ is nondegenerate if its n-th outer power does not vanish (${\omega }^n\ne 0$) anywhere on $M$. In Hamiltonian dynamics $M$ is usually phase space of classical dynamical system with finite numbers of degrees of freedom and the symplectic form $\omega$ is basic object that defines Poisson bracket structure, algebra of Hamiltonian vector fields and the form of Hamilton's equations.

The symplectic form $\omega$ naturally defines isomorphism between vector fields and differential 1-forms on $M$ (in other words tangent bundle $TM$ of symplectic manifold can be quite naturally identified with cotangent bundle $T^{*}M$). The isomorphic map ${\Phi }_{\omega }$ from $TM$ into $T^{*}M$ is obtained by taking contraction of the vector field with $\omega$ $${\Phi }_{\omega }:X\to -i_X\omega$$ (minus sign is the matter of convention). This isomorphism gives rise to natural classification of vector fields. Namely, vector field $X_h$ is said to be Hamiltonian if its image is exact 1-form or in other words if it satisfies Hamilton's equation $$i_{X_h}\omega +dh=0$$ for some function $h$ on $M$. Similarly, vector field $X$ is called locally Hamiltonian if it's image is closed 1-form $$i_X\omega +u=0,du=0$$

One of the nice features of locally Hamiltonian vector fields, known as Liouville's theorem, is that these vector fields preserve symplectic form $\omega$. In other words Lie derivative of the symplectic form $\omega$ along arbitrary locally Hamiltonian vector field vanishes $$L_X\omega =0\iff i_X\omega +du=0,du=0$$ Indeed, using Cartan's formula that expresses Lie derivative in terms of contraction and exterior derivative $$L_X=i_{X}d+d{i}_X$$ one gets $$L_X\omega =i_{X}d\omega +d{i}_X\omega =d{i}_X\omega$$ (since $d\omega =0$) but according to the definition of locally Hamiltonian vector field $$d{i}_X\omega =-du=0$$ So locally Hamiltonian vector fields preserve $\omega$ and vise versa, if vector field preserves symplectic form $\omega$ then it is locally Hamiltonian.

Clearly, Hamiltonian vector fields constitute subset of locally Hamiltonian ones since every exact 1-form is also closed. Moreover one can notice that Hamiltonian vector fields form ideal in algebra of locally Hamiltonian vector fields. This fact can be observed as follows. First of all for arbitrary couple of locally Hamiltonian vector fields $X,Y$ we have $L_X\omega =L_Y\omega =0$ and $$L_X{L}_Y\omega -L_Y{L}_X\omega =L_{[X,Y]}\omega =0$$ so locally Hamiltonian vector fields form Lie algebra (corresponding Lie bracket is ordinary commutator of vector fields). Further it is clear that for arbitrary Hamiltonian vector field $X_h$ and locally Hamiltonian one $Z$ one has $$L_Z\omega =0$$ and $$i_{X_h}\omega +dh=0$$ that implies $$\begin{gathered} L_Z(i_{X_h}\omega +dh)=L_{[Z,X_h]}\omega +i_{X_h}{L}_Z\omega +d{L}_{Z}h \\ =L_{[Z,X_h]}\omega +d{L}_{Z}h=0 \end{gathered}$$ thus commutator $[Z,X_h]$ is Hamiltonian vector field $X_{L_{Z}h}$, or in other words Hamiltonian vector fields form ideal in algebra of locally Hamiltonian vector fields.

Isomorphism ${\Phi }_{\omega }$ can be extended to higher order vector fields and differential forms by linearity and multiplicativity. Namely, $${\Phi }_{\omega }(X\wedge Y)={\Phi }_{\omega }\left(X\right)\wedge {\Phi }_{\omega }\left(Y\right)$$ Since ${\Phi }_{\omega }$ is isomorphism, the symplectic form $\omega$ has unique counter image $W$ known as Poisson bivector field. Property $d\omega =0$ together with non degeneracy implies that bivector field $W$ is also nondegenerate ($W^n\ne 0$) and satisfies condition $$[W,W]=0$$ where bracket $[,]$ known as Schouten bracket or supercommutator, is actually graded extension of ordinary commutator of vector fields to the case of multivector fields, and can be defined by linearity and derivation property $$\begin{gathered} [C_1\wedge C_2\wedge ...\wedge C_n,S_1\wedge S_2\wedge ...\wedge S_n]= \\ (-1{)}^{p+q}[C_p,S_q]\wedge C_1\wedge C_2\wedge ...\wedge {Ĉ}_p\wedge ...\wedge C_n \\ \wedge S_1\wedge S_2\wedge ...\wedge {Ŝ}_q\wedge ...\wedge S_n \end{gathered}$$ where over hat denotes omission of corresponding vector field. In terms of the bivector field $W$ Liouville's theorem mentioned above can be rewritten as follows $$\left[W\right(u),W]=0\iff du=0$$ for each 1-form $u$. It follows from graded Jacoby identity satisfied by Schouten bracket and property $[W,W]=0$ satisfied by Poisson bivector field.

Being counter image of symplectic form, $W$ gives rise to map ${\Phi }_W$, transforming differential 1-forms into vector fields, which is inverted to the map ${\Phi }_{\omega }$ and is defined by $${\Phi }_W:u\to W\left(u\right);{\Phi }_W{\Phi }_{\omega }=id$$ Further we will often use these maps.

In Hamiltonian dynamical systems Poisson bivector field is geometric object that underlies definition of Poisson bracket — kind of Lie bracket on algebra of smooth real functions on phase space. In terms of bivector field $W$ Poisson bracket is defined by $$\{f,g\}=W(df\wedge dg)$$ The condition $[W,W]=0$ satisfied by bivector field ensures that for every triple $(f,g,h)$ of smooth functions on the phase space the Jacobi identity $$\left\{f\right\{g,h\left\}\right\}+\left\{h\right\{f,g\left\}\right\}+\left\{g\right\{h,f\left\}\right\}=0.$$ is satisfied. Interesting property of the Poisson bracket is that map from algebra of real smooth functions on phase space into algebra of Hamiltonian vector fields, defined by Poisson bivector field $$f\to X_f=W\left(df\right)$$ appears to be homomorphism of Lie algebras. In other words commutator of two vector fields associated with two arbitrary functions reproduces vector field associated with Poisson bracket of these functions $$[X_f,X_g]=X_{\{f,g\}}$$ This property is consequence of the Liouville theorem and definition of Poisson bracket. Further we also need another useful property of Hamiltonian vector fields and Poisson bracket $$\{f,g\}=W(df\wedge dg)=\omega (X_f\wedge X_g)=L_{X_f}g=-L_{X_g}g$$ it also follows from Liouville theorem and definition of Hamiltonian vector fields and Poisson brackets.

To define dynamics on $M$ one has to specify time evolution of observables (smooth functions on $M$). In Hamiltonian dynamical systems time evolution is governed by Hamilton's equation $$\frac{d}{dt}f=\{h,f\}$$ where $h$ is some fixed smooth function on the phase space called Hamiltonian. In local coordinate frame $z_b$ bivector field $W$ has the form $$W=W_{bc}\frac{\partial }{\partial z_b}\wedge \frac{\partial }{\partial z_c}$$ and the Hamilton's equation rewritten in terms of local coordinates takes the form $${ż}_b=W_{bc}\frac{\partial h}{\partial z_b}$$

Now let us focus on symmetries of Hamilton's equation (24). Generally speaking, symmetries play very important role in Hamiltonian dynamics due to different reasons. They not only give rise to conservation laws but also often provide very effective solutions to problems that otherwise would be difficult to solve. Here we consider special class of symmetries of Hamilton's equation called non-Noether symmetries. Such a symmetries appear to be closely related to many geometric concepts used in Hamiltonian dynamics including bi-Hamiltonian structures, Frölicher-Nijenhuis operators, Lax pairs and bicomplexes.

Before we proceed let us recall that each vector field $E$ on the phase space generates the one-parameter continuous group of transformations $$g_z=e^{z{L}_E}$$ (here $L$ denotes Lie derivative) that acts on the observables as follows $$g_z\left(f\right)=e^{z{L}_E}\left(f\right)=f+z{L}_{E}f+\mathrm{½}(z{L}_E{)}^{2}f+\cdots$$ Such a group of transformation is called symmetry of Hamilton's equation (24) if it commutes with time evolution operator $$\frac{d}{dt}{g}_z\left(f\right)=g_z\left(\frac{d}{dt}f\right)$$ in terms of the vector fields this condition means that the generator $E$ of the group $g_z$ commutes with the vector field $W\left(h\right)=\{h,\}$, i. e. $$[E,W(h\left)\right]=0.$$ However we would like to consider more general case where $E$ is time dependent vector field on phase space. In this case (30) should be replaced with $$\frac{\partial }{\partial t}E=[E,W(h\left)\right].$$

Further one should distinguish between groups of symmetry transformations generated by Hamiltonian, locally Hamiltonian and non-Hamiltonian vector fields. First kind of symmetries are known as Noether symmetries and are widely used in Hamiltonian dynamics due to their tight connection with conservation laws. Second group of symmetries is rarely used. While third group of symmetries that further will be referred as non-Noether symmetries seems to play important role in integrability issues due to their remarkable relationship with bi-Hamiltonian structures and Frölicher-Nijenhuis operators. Thus if in addition to (30) the vector field $E$ does not preserve Poisson bivector field $[E,W]\ne 0$ then $g_z$ is called non-Noether symmetry.

Now let us focus on non-Noether symmetries. We would like to show that the presence of such a symmetry essentially enriches the geometry of the phase space and under the certain conditions can ensure integrability of the dynamical system. Before we proceed let us recall that the non-Noether symmetry leads to a number of integrals of motion. More precisely the relationship between non-Noether symmetries and the conservation laws is described by the following theorem. This theorem was proposed by Lutzky in [51]. Here it is reformulated in terms of Poisson bivector field.

Besides Hamiltonian dynamical systems that admit invariant symplectic form $\omega$, there are dynamical systems that either are not Hamiltonian or admit Hamiltonian realization but explicit form of symplectic structure $\omega$ is unknown or too complex. However usually such a dynamical systems possess invariant volume form $\Omega$ which like symplectic form can be effectively used in construction of conservation laws. Note that volume form for given manifold is arbitrary differential form of maximal degree (equal to the dimension of manifold). In case of regular Hamiltonian systems, n-th outer power of the symplectic form $\omega$ naturally gives rise to the invariant volume form known as Liouville form $$\Omega ={\omega }^n$$ and sometimes it is easier to work with $\Omega$ rather then with symplectic form itself. In generic Liouville dynamical system time evolution is governed by equations of motion $$\frac{d}{dt}f=X\left(f\right)$$ where $X$ is some smooth vector field that preserves Liouville volume form $\Omega$ $$\frac{d}{dt}\Omega =L_X\Omega =0$$ Symmetry of equations of motion still can be defined by condition $$\frac{d}{dt}{g}_z\left(f\right)=g_z\left(\frac{d}{dt}f\right)$$ that in terms of vector fields implies that generator of symmetry $E$ should commute with time evolution operator $X$ $$[E,X]=0$$ Throughout this chapter symmetry will be called non-Liouville if it is not conformal symmetry of $\Omega$, or in other words if $$L_E\Omega \ne c\Omega$$ for any constant $c$. Such a symmetries may be considered as analog of non-Noether symmetries defined in Hamiltonian systems and similarly to the Hamiltonian case one can try to construct conservation laws by means of generator of symmetry $E$ and invariant differential form $\Omega$. Namely we have the following theorem, which is reformulation of Hojman's theorem in terms of Liouville volume form.

Presence of the non-Noether symmetry not only leads to a sequence of conservation laws, but also endows the phase space with a number of interesting geometric structures and it appears that such a symmetry is related to many important concepts used in theory of dynamical systems. One of the such concepts is Lax pair that plays quite important role in construction of completely integrable models. Let us recall that Lax pair of Hamiltonian system on Poisson manifold $M$ is a pair $(L,P)$ of smooth functions on $M$ with values in some Lie algebra $g$ such that the time evolution of $L$ is given by adjoint action $$\frac{d}{dt}L=[L,P]=-a{d}_{P}L$$ where $[,]$ is a Lie bracket on $g$. It is well known that each Lax pair leads to a number of conservation laws. When $g$ is some matrix Lie algebra the conservation laws are just traces of powers of $L$ $$I^{\left(k\right)}=\frac{1}{2}Tr\left(L^k\right)$$ since trace is invariant under coadjoint action $$\begin{gathered} \frac{d}{dt}{I}^{\left(k\right)}=\frac{1}{2}\frac{d}{dt}Tr\left(L^k\right)=\frac{1}{2}Tr\left(\frac{d}{dt}{L}^k\right)=\frac{k}{2}Tr\left(L^{k-1}\frac{d}{dt}L\right) \\ =\frac{k}{2}Tr\left(L^{k-1}\right[L,P\left]\right)=\frac{1}{2}Tr\left(\right[L^k,P\left]\right)=0 \end{gathered}$$ It is remarkable that each generator of the non-Noether symmetry canonically leads to the Lax pair of a certain type. Such a Lax pairs have definite geometric origin, their Lax matrices are formed by coefficients of invariant tangent valued 1-form on the phase space. In the local coordinates $z_s$, where the bivector field $W$, symplectic form $\omega$ and the generator of the symmetry $E$ have the following form $$W=\sum _{rs}{W}_{rs}\frac{\partial }{\partial z_r}\wedge \frac{\partial }{\partial z_r}\omega =\sum _{rs}{\omega }_{rs}d{z}_r\wedge d{z}_{s}E=\sum _s{E}_s\frac{\partial }{\partial z_s}$$ corresponding Lax pair can be calculated explicitly. Namely we have the following theorem (see also [55]-[56]):

Now let us focus on the integrability issues. We know that $n$ integrals of motion are associated with each generator of non-Noether symmetry, in the same time we know that, according to the Liouville-Arnold theorem, regular Hamiltonian system $(M,h)$ on $2n$ dimensional symplectic manifold $M$ is completely integrable (can be solved completely) if it admits $n$ functionally independent integrals of motion in involution. One can understand functional independence of set of conservation laws $c_1,c_2...c_n$ as linear independence of either differentials of conservation laws $d{c}_1,d{c}_2...d{c}_n$ or corresponding Hamiltonian vector fields $X_{c_1},X_{c_2}...X_{c_n}$. Strictly speaking we can say that conservation laws $c_1,c_2...c_n$ are functionally independent if Lesbegue measure of the set of points of phase space $M$ where differentials $d{c}_1,d{c}_2...d{c}_n$ become linearly dependent is zero. Involutivity of conservation laws means that all possible Poisson brackets of these conservation laws vanish pair wise $$\{c_i,c_j\}=0i,j=1...n$$ In terms of the vector fields, existence of involutive family of $n$ functionally independent conservation laws $c_1,c_2...c_n$ implies that corresponding Hamiltonian vector fields $X_{c_1},X_{c_2}...X_{c_n}$ span Lagrangian subspace (isotropic subspace of dimension $n$) of tangent space (at each point of $M$). Indeed, due to property (23) $$\{c_i,c_j\}=\omega (X_{c_i},X_{c_j})=0$$ thus space spanned by $X_{c_1},X_{c_2}...X_{c_n}$ is isotropic. Dimension of this space is $n$ so it is Lagrangian. Note also that distribution $X_{c_1},X_{c_2}...X_{c_n}$ is integrable since due to (22) $$[X_{c_i},X_{c_j}]=X_{\{c_i,c_j\}}=0$$ and according to Frobenius theorem there exists submanifold of $M$ such that distribution $X_{c_1},X_{c_2}...X_{c_n}$ spans tangent space of this submanifold. Thus for phase space geometry existence of complete involutive set of integrals of motion implies existence of invariant Lagrangian submanifold.

Now let us look at conservation laws $Y^{\left(1\right)},Y^{\left(2\right)}...Y^{\left(n\right)}$ associated with generator of non-Noether symmetry. Generally speaking these conservation laws might appear to be neither functionally independent nor involutive. However it is reasonable to ask the question – what condition should be satisfied by the generator of the non-Noether symmetry to ensure the involutivity ($\{Y^{\left(k\right)},Y^{\left(m\right)}\}=0$) of conserved quantities? In Lax theory situation is very similar — each Lax matrix leads to the set of conservation laws but in general this set is not involutive, however in Lax theory there is certain condition known as Classical Yang-Baxter Equation (CYBE) that being satisfied by Lax matrix ensures that conservation laws are in involution. Since involutivity of the conservation laws is closely related to the integrability, it is essential to have some analog of CYBE for the generator of non-Noether symmetry. To address this issue we would like to propose the following theorem.

Further we will focus on non-Noether symmetries that satisfy condition (110). Besides yielding involutive families of conservation laws, such a symmetries appear to be related to many known geometric structures such as bi-Hamiltonian systems [53] and Frölicher-Nijenhuis operators (torsionless tangent valued differential 1-forms). The relationship between non-Noether symmetries and bi-Hamiltonian structures was already implicitly outlined in the proof of Theorem 4. Now let us pay more attention to this issue.

Originally bi-Hamiltonian structures were introduced by F. Magri in analisys of integrable infinite dimensional Hamiltonian systems such as Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) hierarchies, Nonlinear Schrödinger equation and Harry Dym equation. Since that time bi-Hamiltonian formalism is effectively used in construction of involutive families of conservation laws in integrable models

Generic bi-Hamiltonian structure on $2n$ dimensional manifold consists out of two Poisson bivector fields $W$ and $Ŵ$ satisfying certain compatibility condition $[Ŵ,W]=0$. If, in addition, one of these bivector fields is nondegenerate ($W^n\ne 0$) then bi-Hamiltonian system is called regular. Further we will discuss only regular bi-Hamiltonian systems. Note that each Poisson bivector field by definition satisfies condition (15). So we actually impose four restrictions on bivector fields $W$ and $Ŵ$ $$[W,W]=[Ŵ,W]=[Ŵ,Ŵ]=0$$ and $$W^n\ne 0$$ During the proof of Theorem 4 we already showed that bivector fields $W$ and $Ŵ=[E,W]$ satisfy conditions (126) (see (112)-(116)), thus we can formulate the following statement

In terms of differential forms bi-Hamiltonian structure is formed by couple of closed differential 2-forms: symplectic form $\omega$ (such that $d\omega =0$ and ${\omega }^n\ne 0$) and ${\omega }^{\ast }=L_E\omega$ (clearly $d{\omega }^{\ast }=d{L}_E\omega =L_{E}d\omega =0$). It is important that by taking Lie derivative of Hamilton's equation $$i_{X_h}\omega +dh=0$$ along the generator $E$ of symmetry $$L_E(i_{X_h}\omega +dh)=i_{[E,X_h]}\omega +i_{X_h}{L}_E\omega +L_{E}dh=i_{X_h}{\omega }^{\ast }+d{L}_{E}h=0$$ one obtains another Hamilton's equation $$i_{X_h}{\omega }^{\ast }+d{h}^{\ast }=0$$ where $h^{\ast }=L_{E}h$. This is actually second Hamiltonian realization of equations of motion and thus under certain conditions existence of non-Noether symmetry gives rise to additional presymplectic structure ${\omega }^{\ast }$ and additional Hamiltonian realization of the dynamical system. In many integrable models admitting bi-Hamiltonian realization (including Toda chain, Korteweg-de Vries hierarchy, Nonlinear Schrödinger equation, Broer-Kaup system and Benney system) non-Noether symmetries that are responsible for existence of bi-Hamiltonian structures has been found and motivated further investigation of relationship between symmetries and bi-Hamiltonian structures. Namely it seems to be interesting to know whether in general case existence of bi-Hamiltonian structure is related to non-Noether symmetry. Let us consider more general case and suppose that we have couple of differential 2-forms $\omega$ and ${\omega }^{\ast }$ such that $$d\omega =d{\omega }^{\ast }=0,{\omega }^n\ne 0$$ $$i_{X_h}\omega +dh=0$$ and $$i_{X_h}{\omega }^{\ast }+d{h}^{\ast }=0$$ The question is whether there exists vector field $E$ (generator of non-Noether symmetry) such that $[E,X_h]=0$ and ${\omega }^{\ast }=L_E\omega$.

The answer depends on ${\omega }^{\ast }$. Namely if ${\omega }^{\ast }$ is exact form (there exists 1-form ${\theta }^{\ast }$ such that ${\omega }^{\ast }=d{\theta }^{\ast }$) then one can argue that such a vector field exists and thus any exact bi-Hamiltonian structure is related to hidden non-Noether symmetry. To outline proof of this statement let us introduce vector field $E^{\ast }$ defined by $$i_{E^{\ast }}\omega ={\theta }^{\ast }$$ (such a vector field always exist because $\omega$ is nondegenerate 2-form). By construction $$L_{E^{\ast }}\omega ={\omega }^{\ast }$$ Indeed $$L_{E^{\ast }}\omega =d{i}_{E^{\ast }}\omega +i_{E^{\ast }}d\omega =d{\theta }^{\ast }={\omega }^{\ast }$$ And $$i_{[E^{\ast },X_h]}\omega =L_{E^{\ast }}\left(i_{X_h}\omega \right)-i_{X_h}{L}_{E^{\ast }}\omega =-d\left(E^{\ast }\right(h)-h^{\ast })=-dh\text{'}$$ In other words $[X_h,E^{\ast }]$ is Hamiltonian vector field $$[X_h,E]=X_{h\text{'}}$$ One can also construct locally Hamiltonian vector field $X_g$, that satisfies the same commutation relation. Namely let us define function (in general case this can be done only locally) $$g\left(z\right)=\underset{0}{\overset{t}{\int }}h\text{'}dt$$ where integration along solution of Hamilton's equation, with fixed origin and end point in $z\left(t\right)=z$, is assumed. And then it is easy to verify that locally Hamiltonian vector field associated with $g\left(z\right)$, by construction, satisfies the same commutation relations as $E^{\ast }$ (namely $[X_h,X_g]=X_{h\text{'}}$). Using $E^{\ast }$ and $X_{h\text{'}}$ one can construct generator of non-Noether symmetry — non-Hamiltonian vector field $E=E^{\ast }-X_g$ commuting with $X_h$ and satisfying $$L_E\omega =L_{E^{\ast }}\omega -L_{X_g}\omega =L_{E^{\ast }}\omega ={\omega }^{\ast }$$ (thanks to Liouville's theorem $L_{X_g}\omega =0$). So in case of regular Hamiltonian system every exact bi-Hamiltonian structure is naturally associated with some (non-Noether) symmetry of space of solutions. In case where bi-Hamiltonian structure is not exact (${\omega }^{\ast }$ is closed but not exact) then due to $${\omega }^{\ast }=L_E\omega =d{i}_E\omega +i_{E}d\omega =d{i}_E\omega$$ it is clear that such a bi-Hamiltonian system is not related to symmetry. However in all known cases bi-Hamiltonian structures seem to be exact.

Another important concept that is often used in theory of dynamical systems and may be related to the non-Noether symmetry is the bidifferential calculus (bicomplex approach). Recently A. Dimakis and F. Müller-Hoissen applied bidifferential calculi to the wide range of integrable models including KdV hierarchy, KP equation, self-dual Yang-Mills equation, Sine-Gordon equation, Toda models, non-linear Schrödinger and Liouville equations. It turns out that these models can be effectively described and analyzed using the bidifferential calculi [17], [24]. Here we would like to show that each generator of non-Noether symmetry satisfying condition $\left[\right[E[E,W]\left]W\right]=0$ gives rise to certain bidifferential calculus.

Before we proceed let us specify what kind of bidifferential calculi we plan to consider. Under the bidifferential calculus we mean the graded algebra of differential forms over the phase space $$\Omega =\underset{k=0}{\overset{\mathrm{\infty }}{\cup }}{\Omega }^{\left(k\right)}$$ (${\Omega }^{\left(k\right)}$ denotes the space of $k$-degree differential forms) equipped with a couple of differential operators $$d,\text{đ}:{\Omega }^{\left(k\right)}\to {\Omega }^{(k+1)}$$ satisfying conditions $d^2={\text{đ}}^2=d\text{đ}+\text{đ}d=0$ (see [24]). In other words we have two De Rham complexes $M,\Omega ,d$ and $M,\Omega ,\text{đ}$ on algebra of differential forms over the phase space. And these complexes satisfy certain compatibility condition — their differentials anticommute with each other $d\text{đ}+\text{đ}d=0$. Now let us focus on non-Noether symmetries. It is interesting that if generator of the non-Noether symmetry satisfies equation $\left[\right[E[E,W]\left]W\right]=0$ then we are able to construct an invariant bidifferential calculus of a certain type. This construction is summarized in the following theorem: